Bi-modal Godel logic over [0,1]-valued

Kripke frames

Xavier Caicedo ∗ Ricardo Oscar Rodrıguez †

Abstract

We consider the Godel bi-modal logic determined by fuzzy Kripkemodels where both the propositions and the accessibility relation areinfinitely valued over the standard Godel algebra [0,1], and prove strongcompleteness of Fischer Servi intuitionistic modal logic IK plus theprelinearity axiom with respect to this semantics. We axiomatize alsothe bi-modal analogues of classical T , S4, and S5, obtained by re-stricting to models over frames satisfying the [0,1]-valued versions ofthe structural properties which characterize these logics. As applica-tion of the completeness theorems we obtain a representation theoremfor bi-modal Godel algebras.

In a previous paper [6], we have considered a semantics for Godel modallogic based on fuzzy Kripke models where both the propositions and theaccessibility relation take values in the standard Godel algebra [0,1], wecall these Godel-Kripke models, and we have provided strongly completeaxiomatizations for the uni-modal fragments of this logic with respect tovalidity and semantic entailment from countable theories. The systems G�and G3 axiomatizing the �-fragment and the 3-fragment, respectively, areobtained by adding to Godel-Dummett propositional calculus the followingaxiom schemes and inference rules:

G�: �(φ→ ψ)→ (�φ→ �ψ)¬¬�φ→ �¬¬φFrom φ, infer �φ

G3: 3(φ ∨ ψ)→ (3φ ∨3ψ)3¬¬φ→ ¬¬3φ¬3⊥From φ→ ψ, infer 3φ→ 3ψ.

∗Departamento de Matematicas, Universidad de los Andes, Bogota, Colombia;xcaicedo@uniandes.edu.co

†Departamento de Computacion, Fac. Ciencias Exactas y Naturales, Universi-dad de Buenos Aires, 1428 Buenos Aires, Argentina; ricardo@dc.uba.ar

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These two logics diverge substantially in their model theoretic properties.Thus, G� does not have the finite model property while G3 does, and thefirst logic is characterized by models with {0,1}-valued accessibility relation(accessibility-crisp models) while the second one does not. Similar resultswere obtained for the uni-modal Godel analogues of the classical modal log-ics T and S4 determined by Godel-Kripke models over frames satisfying,respectively, the [0,1]-valued version of reflexivity, or reflexivity and transi-tivity. The axiomatization of the uni-modal Godel analogues of S5 remainsopen.

It is the main purpose of this paper to show that the full bi-modallogic based in Godel-Kripke models is axiomatized by the system G�3 whichresults by adding to the union of G� and G3 Fischer Servi’s connectingaxioms [14]:

3(φ→ ψ)→ (2φ→ 3ψ)(3φ→ 2ψ)→ 2(φ→ ψ),

and to extend this completeness result to the bi-modal Godel analogues ofclassical T, S4, S5, and related systems.

We discuss briefly at the end of the paper an embedding of our semanticsinto algebraic semantics for G�3 and its extensions, and utilize our com-pleteness theorem to show a functional representation theorem for bi-modalGodel algebras.

The many valued Kripke interpretation of bi-modal logic utilized in thispaper was proposed originally by Fitting [15], [16], with a complete Heytingalgebra as algebra of truth values, and he gave a complete axiomatizationassuming the algebra was finite and the language had constants for all thetruth values. See also [21] and [11]. Bou, Esteva, and Godo [5] have proposedutilizing this kind of interpretation for general algebras in the study of fuzzymodal logics. Our methods of proof do not seem to extend easily, however,to algebras distinct from the Godel algebra [0,1], and we do not know anyother completeness result for this type of semantics for a fixed algebra H,except Fitting’s result quoted above and Metcalfe & Olivetti completenessof natural deduction systems for G� and G3 [22].G�3 results equivalent to the system IK, proposed by Fischer-Servi [14]

as the intuitionistic counterpart of classical modal logic K, plus the prelin-earity axiom: (φ→ ψ)∨(ψ → φ). Similarly, the Godel analogue of bi-modalS5 results equivalent to the system MIPC of Bull [4] and Prior [26] plusprelinearity.

IK and its extensions have been extensively studied, either by means ofclassical Kripke models for intuitionism equipped with extra relations com-

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muting with the order to interpret the modal operators ([28], [24], [25], [10],[29], [30], [17], [7], [9]), or by means of algebraic interpretations, speciallyin the case of MIPC, known to be complete for values in monadic Heytingalgebras ([4], [24], [13], [1], [2]). A major result is that both logics enjoy thefinite model property under these semantics. Clearly, G�3 and its modalextensions inherit similar semantics, but those interpretations do not havethe standard character of Godel-Kripke semantics relevant to fuzzy logic,and it does not seem possible to derivate our results from their properties.For example, the formula �¬¬θ → ¬¬�θ has finite counter-models withrespect to those semantics but not in Godel-Kripke semantics.

1 Godel-Kripke models

The language L�3(V ar) of propositional bi-modal logic is built from a setV ar of propositional variables, connectives symbols ∨,∧,→,⊥, and themodal operators symbols � and 3. Other connectives are defined as usual:⊤ := φ→ φ, ¬φ := φ→ ⊥, φ←→ ψ := (φ→ ψ) ∧ (ψ → φ). We will writeL�3 if the set V ar is understood.

Recall that a linear Heyting algebra, or Godel algebra in the fuzzy litera-ture, is a Heyting algebra satisfying the identity (x⇒ y)g(y ⇒ x) = 1. Thevariety of these algebras is generated by the standard Godel algebra [0, 1],the ordered interval with its unique Heyting algebra structure. Let the sym-bols ·, ⇒, g, denote, respectively, the meet, residuum (implication), andjoin operations of [0, 1]. For convenience, we take g as primitive although itis definable in Godel algebras as xg y = ((x⇒ y)⇒ y)) · ((y ⇒ x)⇒ x)).

Definition 1.1 A Godel-Kripke model (GK-model) will be a structureM = ⟨W,S, e⟩ where W is a non-empty set of objects that we call worlds ofM, and S : W ×W → [0, 1], e : W × V ar → [0, 1] are arbitrary functions.The pair ⟨W,S⟩ will be called a GK-frame.

The function e : W × V ar → [0, 1] associates to each world x a valuatione(x,−) : V ar → [0, 1] which extends to e(x,−) : L�3(V ar) → [0, 1] bydefining inductively on the construction of the formulas (we utilize the samesymbol e to name the extension):

e(x,⊥) := 0e(x, φ ∧ ψ) := e(x, φ) · e(x, ψ)e(x, φ ∨ ψ) := e(x, φ)g e(x, ψ)e(x, φ→ ψ) := e(x, φ)⇒ e(x, ψ)

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e(x,2φ) := infy∈W {Sxy ⇒ e(y, φ)}e(x,3φ) := supy∈W {Sxy · e(y, φ)}.

Truth, validity and entailment are defined for φ ∈ L�3, T ⊆ L�3 as follows:

φ is true in M at x, written M |=x φ, if e(x, φ) = 1.φ is valid in M, written M |= φ, if M |=x φ at any world x of M.φ is GK-valid , written |=GK φ, if M |= φ for any GK-model M .T |=GK φ if and only if for any GK-model M and any world x in M :

M |=x θ for all θ ∈ T implies M |=x φ.

It is routine to verify that all axiom schemes corresponding to identitiessatisfied in [0, 1]; that is, the laws of Godel-Dummett logic, are GK-valid.In addition

Proposition 1.1 The following schemes are GK-valid:

(K�) �(φ→ ψ)→ (�φ→ �ψ)(K3) 3(φ ∨ ψ)→ (3φ ∨3ψ)(F3) ¬3⊥(FS1) 3(φ→ ψ)→ (2φ→ 3ψ)(FS2) (3φ→ 2ψ)→ 2(φ→ ψ).

Proof: Let M = ⟨W,S, e⟩ be a GK-model. (K�): By definition and prop-erties of the residuum, e(x,�(φ → ψ)) · e(x,�φ) ≤ (Sxy ⇒ (e(y, φ) ⇒e(y, ψ)) · (Sxy ⇒ e(y, φ)) ≤ (Sxy ⇒ e(y, ψ)) for any y ∈ W . Taking themeet over y in the last expression: e(x,�(φ → ψ)) · e(x,�φ) ≤ e(x,�ψ),hence e(x,�(φ → ψ)) ≤ e(x,�φ → �ψ). (K3): By distributivity andproperties of the join: e(3(x, φ ∨ ψ)) = supy{Sxy · (e(y, φ) g e(y, ψ))} =supy{Sxy·e(y, φ)}gsupy{Sxy·e(y, ψ)}. (F3): e(x,3⊥) = supy{Sxy·0} = 0.(FS1): Sxy · e(x,2φ) · e(y, φ → ψ) ≤ Sxy · (Sxy ⇒ e(y, φ)) · (e(y, φ) ⇒e(y, ψ)) ≤ Sxy · e(y, ψ)) ≤ e(x,3ψ). Therefore, Sxy · e(y, φ → ψ) ≤(e(x,2φ) ⇒ e(x,3ψ)), and taking the join over y in the left hand side,we have e(x,3(φ → ψ)) ≤ e(x,2φ → 3ψ). (FS2): e(x,3φ → 2ψ) ≤[Sxy · e(y, φ) ⇒ (Sxy ⇒ e(y, ψ))] = [Sxy · e(y, φ) ⇒ e(y, ψ)] = (Sxy ⇒e(y, φ→ ψ)). �

Remark. Utilizing any complete Heyting algebra H instead of [0, 1] inthe above definitions, we obtain H-valued Kripke models (HK-models) andcorresponding notions of HK-validity and entailment. Then the laws inProposition 1.1 are HK-valid, as are the laws of the intermediate proposi-tional logic determined by H.

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2 A bi-modal calculus

Let G be some axiomatic version of Godel-Dummett propositional calculus;that is, Heyting calculus plus the axiom (φ → ψ) ∨ (ψ → φ), and let ⊢Gdenote deduction in this logic. Let L(X) denote the set of formulas built bymeans of the connectives ∧,→, and ⊥, from a given set X. For simplicity,the extension of a valuation v : X → [0, 1] to L(X) according to the Heytinginterpretation of the connectives will be denoted v also. It is well known thatthis system is complete for validity with respect to these valuations and thedistinguished value 1. We will need the fact that it is actually sound andcomplete in the following stronger sense (see [6]):

Proposition 2.1 i) If T∪{φ} ⊆ L(X), then T ⊢G φ implies inf v(T ) ≤ v(φ)for any valuation v : X → [0, 1]. ii) If T is countable, and T 0G φi1 ∨ ..∨φi1for each finite subset of a countable family {φi}i there is a valuation v : L→[0, 1] such that v(θ) = 1 for all θ ∈ T and v(φi) < 1 for all i.

For an example that completeness for [0,1]-valued entailment can not beextended to uncountable theories see Section 3 in [6] and also Proposition3.1 below.

Definition 2.1 G�3 is the deductive calculus obtained by adding to G theschemes of Proposition 1.1 and the inference rules:

(NR�) From φ infer �φ(RN3) From φ→ ψ infer 3φ→ 3ψ.

Proofs with assumptions are allowed with the restriction that NR� andRN3 may be applied only when the premise is a theorem; ⊢G�3

will denotededuction in this system.

The restriction on the application of the rules allows the Deduction The-orem that we will utilize freely without quoting it:

Lemma 2.1 T, ψ ⊢G23 φ implies T ⊢G23 ψ → φ.

An alternative axiomatization of G�3 is obtained by replacing FS1 withthe scheme

(P) �(φ→ ψ)→ (3φ→ 3ψ)

and deleting the rule RN3.Indeed, ⊢G23 3φ → 3((φ → ψ) → ψ) ⊢G23 3φ → (�(φ → ψ) → 3ψ)

⊢G23 �(φ → ψ) → (3φ → 3ψ) by Heyting calculus, RN3, and FS1. On

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the other hand, P + NR� deduce trivially RN3, and FS1 is deduced from(G23r{FS1,RN3})+{P} as follows: ⊢ �φ→ �((φ→ ψ)→ ψ) by Heytingcalculus plus NR� and K�; thus ⊢ �φ → (3(φ → ψ) → 3ψ) by P, and⊢ 3(φ→ ψ)→ (�φ→ 3ψ) by Heyting calculus.

Theorem 2.1 (Soundness) T ⊢G23 φ implies T |=GK φ.

Proof: Clearly, the Modus Ponens rule preserves truth at every world of anyGK-model M , and the rule NR� preserves validity (truth at all worlds) inany model since M |= φ implies M |= �φ, trivially. Similarly, M |= φ→ ψimplies Sxy · e(y, φ) ≤ Sxy · e(y, ψ) for all x, y, and thus M |= 3φ → 3ψ.The rest follows from Proposition 1.1. �

It is easy to provide counterexamples to the validity of ¬�¬θ → 3θand ¬3¬θ → �θ; thus the modal operators are not interdefinable in G�3

in the classical way. In fact, they are not interdefinable in any manner. Forexample, the invalid formula �¬¬θ → ¬¬�θ is not expressible in termsof 3 alone because the 3-fragment has the finite model property for GK-semantics with respect to the number of worlds, while this formula has nofinite counterexample (cf. [6]).

The following are some theorems of G23. The first one is an axiom inFitting’s systems in [15], the next two show the fact claimed in the intro-duction that G�3 is the union of G� and G3 plus the Fischer Servi axioms,the fourth one will be useful in our completeness proof and is the only onedepending on prelinearity.

T1. ¬3θ ←→ �¬θT2. ¬¬�θ → �¬¬θT3. 3¬¬φ→ ¬¬3φT4. (�φ→ 3ψ) ∨�((φ→ ψ)→ ψ)

To see this, write temporarily ⊢ for ⊢G23 . (T1) ¬3θ ⊢ (3θ → �⊥) ⊢�(θ → ⊥) by Heyting calculus and FS2. Similarly, 3θ ⊢ 3(¬θ → ⊥)⊢ �¬θ → 3⊥ ⊢ ¬�¬θ by Heyting calculus, RN3, and FS2; thus, �¬θ ⊢¬3θ by Heyting calculus. (T2) (�φ → ⊥) → ⊥ ⊢ (�φ → 3⊥) → ⊥⊢ 3(φ → ⊥) → �⊥ ⊢ �((φ → ⊥) → ⊥) by F3, FS2, and FS1. (T3)3(¬φ→ ⊥) ⊢ (2¬φ→ 3⊥) ⊢ (¬3φ→ ⊥) by FS1, T1, and F3. (T4) Byprelinearity: ⊢ (�φ→ 3(φ→ ψ)) ∨ (3(φ→ ψ)→ �φ), but �φ→ 3(φ→ψ) ⊢ �φ → (�φ → 3ψ) ⊢ �φ → 3ψ by FS1; moreover, 3(φ → ψ) → �φ⊢ �((φ→ ψ)→ φ) ⊢ �((φ→ ψ)→ ψ) by FS2, Heyting calculus, and RN�.

To prove completeness of ⊢G23 , we will utilize the following convenientreduction of G23 to pure Godel calculus:

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Lemma 2.2 Let ThG�3 be the set of theorems of G�3 with no assumptions,then for any theory T and formula φ in L23 : T ⊢G23 φ if and only ifT ∪ ThG�3 ⊢G φ.

Proof: The rules NR�, RN3 are applied only to formulas in ThG�3, andthis set is closed under those rules. �Remark. G�3 is essentially Fischer Servi system IK ([14], [28]) plus theprelinearity axiom. Moreover, T ⊢IK φ implies T |=HK φ for any completeH. This provides a new interpretation of IK under which one would expectthis logic to be complete.

3 Completeness

In this section, we prove strong completeness of G�3 with respect to entail-ment from countable theories in Godel-Kripke semantics.

We will obtain a finer result for theories T ⊆ L�3 closed under the ruleNR� (T ⊢G�3

θ implies T ⊢G�3�θ). Call these theories normal. It follows

from the observation on an alternative axiomatization in the previous sectionthat a normal theory is also closed under the rule RN3. Clearly, the emptytheory is normal.

Our strategy is to show first completeness for entailment from finitetheories (weak completeness), and utilize a first order compactness argumentto lift this to countable theories. To achieve the first goal, we define foreach normal theory Σ and finite fragment F ⊆ L�3 (a subset closed undersubformulas and containing the formula ⊥) a canonical modelMΣ,F in whichΣ ∩ F will be valid.

Let X := {�θ,3θ : θ ∈ L�3} be the set of formulas in L�3 beginningwith a modal operator; then L�3(V ar) = L(V ar∪X). That is, any formulain L�3(V ar) may be seen as Heyting calculus formula built from the set ofpropositional variables V ar∪X. The canonical model MΣ,F = (WΣ, SF , eF )is defined as follows:

• WΣ is the set of valuations v ∈ [0, 1]V ar∪X such that v(Σ ∪ ThG�3) = 1,where Σ ∪+ThG�3 is considered a subset of L(V ar ∪X).• SF vw = infψ∈F {(v(2ψ)→ w(ψ)) · (w(ψ)→ v(3ψ))}.• eF (v, p) = v(p) for any p ∈ V ar.

Weak completeness will follow from the following lemma which has arather involved proof.

Lemma 3.1 eF (v, φ) = v(φ) for any φ ∈ F and any v ∈WΣ.

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Proof: For simplicity, write W for WΣ. We prove the identity by induc-tion on the complexity of the formulas in F, considered now elements ofL�3(V ar). For ⊥ and the propositional variables in F the equation holdsby definition. The only non trivial inductive steps are: eF (v,2φ) = v(2φ)and eF (v,3φ) = v(3φ) for 2φ,3φ ∈ F. By the inductive hypothesis wemay assume that eF (v′, φ) = v′(φ) for every v′ ∈W ; thus we must prove

infv′∈W

{SF vv′ ⇒ v′(φ)} = v(2φ)) (1)

supv′∈W

{SF vv′ · v′(φ)} = v(3φ)) (2)

By definition, SF vv′ ≤ (v(2φ) ⇒ v′(φ)) and SF vv′ ≤ (v′(φ) ⇒ v(3φ))for any φ ∈ F and v′ ∈ W ; therefore, v(2φ) ≤ (SF vv′ ⇒ v′(φ)) andSF vv′ · v′(φ) ≤ v(3φ). Taking the meet over v′ in the first inequality andthe join in the second,

v(2φ) ≤ infv′∈W

{SF vv′ ⇒ v′(φ)}, supv′∈W

{SF vv′ · v′(φ)} ≤ v(3φ).

Hence, if v(2φ) = 1 and v(3φ) = 0 we obtain (1) and (2), respectively.Therefore, it only remains to prove the next two claims for 2φ,3φ ∈ F .

Claim 1. If v(2φ) = α < 1 and ε > 0, there exists a valuation w ∈ Wsuch that SF vw > w(φ) and w(φ) < α+ ε (thus, (SF vw ⇒ w(φ)) < α+ ε).

Claim 2. If v(3φ) = α > 0 then, for any ε > 0, there exists w ∈ W suchthat w(φ) = 1 and SF vw ≥ α− ε (thus w(φ) · SF vw ≥ α− ε).

Proof of Claim1. By definition of⇒ in [0,1], to grant the required conditionson w it is necessary to find w ∈W and p0 such that α+ε ≥ p0 > w(φ) and forany θ ∈ F : v(2θ) ≤ w(θ) if w(θ) < p0, w(θ) ≤ v(3θ) if v(3θ) < p0. This isachieved in two stages: first producing a valuation u ∈W satisfying u(φ) < 1and the relative ordering conditions the w(θ) must satisfy, conditions whichmay be coded by a theory Γφ,v, and then moving the values u(θ), θ ∈ F,to the correct valuation w by composing u with an increasing bijection of[0,1]. Assume v(2φ) = α < 1 and define (all formulas involved ranging inL�3(V ar))

Γφ,v = {θ : v(2θ) > α} ∪ {θ1 → θ2 : v(3θ1) ≤ v(�θ2)}∪{(θ2 → θ1)→ θ1 : v(3θ1) < v(�θ2)}.

Then we have v(�ξ) > α for each ξ ∈ Γφ,v: for the first set of formulas byconstruction, for the second because v(�(θ1 → θ2)) ≥ v(3θ1 → �θ2) = 1

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by FS2, and for the third, because v(�θ2 → 3θ1) < 1 and thus v(�((θ2 →θ1)→ θ1)) = 1 by T4. This implies that

Γφ,v,Σ ⊢G�3φ.

Otherwise ξ1, . . . , ξk ∈ Γφ,v would exist such that ξ1, . . . , ξk,Σ ⊢G�3φ.

Hence, 2ξ1, . . . ,2ξk,�Σ ⊢G�32φ by NR� and K�,but Σ ⊢G�3

�Σ bynormality. Then 2ξ1, . . . ,2ξk,Σ, ThG�3 ⊢G 2φ by Lemma 2.2 and thus byProposition 2.1 (i), and recalling that v(Σ ∪ ThG�3) = 1,

α < inf v({2ξ1, . . . ,2ξk} ∪ Σ ∪ ThG�3) ≤ v(�φ) = α,

a contradiction. Therefore, by Proposition 2.1 (ii) there exists a valuationu : V ar∪X 7→ [0, 1] such that u(Γφ,v ∪Σ∪ThG�3) = 1 and u(φ) < 1. Thisimplies the following relations between v and u, that we list for further use(see Figure 1). Given θ1, θ2, θ3,

#1. If v(2θ) > α then u(θ) = 1 (since then θ ∈ Γφ,v)#2 If v(3θ1) ≤ v(�θ2) then u(θ1) ≤ u(θ2) (since then θ1 → θ2 ∈ Γφ,v)#3 If v(3θ1) < v(�θ2) then u(θ1) < u(θ2) or u(θ1) = u(θ2) = 1 (since then(θ2 → θ1)→ θ1) ∈ Γφ,v)#4. If v(2θ2) > 0 then u(θ2) > 0 (making θ1 := ⊥ in #3 since u(⊥) =v(3⊥) = 0).

For the next construction we need the finiteness of F. Set B = {v(�θ) :θ ∈ F}, for each b ∈ B define

ub = min{u(θ) : θ ∈ F and v(�θ) = b},

and then define a strictly descending sequence b0, b1, ..., bN = 0 in B asfollows:

b0 = αbi+1 = max{b ∈ B : b < bi and ub < ubi}.

Pick formulas φi ∈ F such that bi = v(�φi) and ubi = u(φi). By finitenessof B, the inductive definition ends with some bN (which could be b0 incase uα = 0). To check that bN = 0, assume bN = v(�φN ) > 0, thenubN = u(φN ) > 0 by property #4 above. But v(�⊥) ≤ v(�φN ) by RN�and K� and u(⊥) = 0. Thus, by minimality of ubN we can not have equality;hence, v(�⊥) < v(�φN ) and thus there exists bN+1 < bN , a contradiction.

By construction, the sequence ub0 , ub1 , ... is also strictly descending withub0 = uα ≤ u(φ) < 1, and it ends at 0 because v(�⊥) ≤ v(�φN ) = 0 andthus ubN ≤ u(⊥) = 0 by minimality again.

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Figure 1: First Translation

Fix ε > 0 such that α+ ε < 1 and further define (taking min ∅ = 1)

p0 = (α+ ε) ·min{v(3θ) : θ ∈ F, α < v(3θ)}pi+1 = bi ·min{v(3θ) : θ ∈ F, bi+1 < v(3θ)} for i ≥ 1.

Notice that we have pi > bi by construction.Summing up,

1 > α+ ε ≥ p0 > b0 = α ≥ p1 > b1 ≥ .... ≥ pN > bN = 0.

1 > ub0 > ub1 > ... > ubN = 0

Now pick an strictly increasing function g : [0, 1] 7→ [0, 1] such that (seeFigure 2)

g(1) = 1g[[uα, 1)] = [α, p0)g[[ubi+1

, ubi)] = [bi+1, pi+1)

Then the valuation w = g ◦ u satisfies w(Σ∪ ThG�3) = 1, and so it belongsto W. Moreover, w(φ) = g(u(φ)) < p0 ≤ α + ε. It remains to show thatSF vw > w(φ). For any θ ∈ F :

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Figure 2: Second Translation

i) If u(θ) = 1 then w(θ) = 1 by definition of w; hence, v(�θ) ≤ w(θ).In addition, v(3θ) ≥ p0, otherwise v(3θ) ≤ α = v(�φ0) which would implyu(θ) ≤ u(φ0) < 1 by #2, a contradiction.

ii) If u(θ) ∈ [ubi , ubi−1) or u(θ) = [ub0 , 1) then v(�θ) ≤ w(θ) ≤ v(3θ).

To see this notice first that w(θ) ∈ [bi, pi) by definition of g. Now, for i ≥ 1,bi is the maximum v(�ψ) with u(ψ) < ubi−1

. Therefore, v(�θ) ≤ bi ≤ w(θ).In addition, for i = 0, v(�θ) ≤ α = b0 ≤ w(θ) by #1. Moreover, if u(θ) =ubi = u(φi) then w(θ) = bi = v(�φi) ≤ v(3θ) by the counter-reciprocal of#3 because ubi < 1, and if u(θ) > ubi then v(3θ) > v(�φi) = bi by thecounter-reciprocal of #2; hence, v(3θ) ≥ pi > w(θ).

It follows form from (i,ii) that infθ∈F {v(�θ) ⇒ w(θ)} = 1 andinfθ∈F {w(θ)⇒ v(3θ)} ≥ p0. Hence, SF vw ≥ p0 > w(φ).

Proof of Claim 2. Again we code first in a relative consistence situation theminimal requirements for w, to obtain u ∈W satisfying those requirements,and then transform u to the correct w by an automorphism of [0,1]. Assume

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v(3φ) = α > 0 and define

Uφ,v = {θ : v(3θ) < α}∪{ϑ2 → ϑ1 : v(3ϑ1) < v(�ϑ2) and v(3ϑ1) < α}∪{(ϑ1 → ϑ2)→ ϑ1 : v(3ϑ1) = v(�ϑ2) and v(3ϑ1) < α}.

This set is non-empty because v(3⊥) = 0. Moreover, for any ξ ∈ Uφ,v wehave v(3ξ) < α; for the first set of axioms by construction; for the secondbecause v(3(ϑ2 → ϑ1)) ≤ v(�ϑ2 → 3ϑ1) = v(3ϑ1) < α by FS1; andfor the third because v(3((ϑ1 → ϑ2) → ϑ1))) ≤ v(�(ϑ1 → ϑ2) → 3ϑ1)≤ v((3ϑ1 → �ϑ2)→ 3ϑ1) = v(3ϑ1) < α by FS1, FS2.

We claim that for any finite {ξ1, . . . , ξk} ⊆ Uφ,v :

φ,Σ ⊢G23ξ1 ∨ . . . ∨ ξk

because, on the contrary, Σ ⊢G23 3φ → 3(ξ1 ∨ . . . ∨ ξk) ⊢G23 3φ →(3ξ1 ∨ . . . ∨3ξk) by normality of Σ and K3, whence

3φ,Σ, ThG�3 ⊢ 3ξ1 ∨ . . . ∨3ξk,

and evaluating with v it would give: α = inf v({3φ} ∪ Σ ∪ ThG�3) ≤max{v(3ξ1), . . . , v(3ξk)} < α, absurd.

Therefore, there is a valuation u such that u(φ) = u(Σ∪TG23) = 1 andu(ξ) < 1 for each ξ ∈ Uφ,v, which has the following consequences for anyθ, θ1, θ2:

##1. If v(3θ) < α then u(θ) < 1 (because then θ ∈ Uφ,.v)##2. If v(3θ1) < v(2θ2) and v(3θ1) < α then u(θ1) < u(θ2) (becauseθ2 → θ1 ∈ Uφ,.v)##3. If v(3θ1) ≤ v(�θ2) and v(3θ1) < α then u(θ1) ≤ u(θ2) (because(θ1 → θ2)→ θ1 ∈ Uφ,v)##4 If u(θ2) = 0 then v(2θ2) = 0 (making θ1 := ⊥ in ##2 and takingcounter-reciprocal)##5. If v(3θ1) = 0 then u(θ) = 0 (making θ2 := ⊥ in ##3, because thenv(3θ1) ≤ v(�⊥) and v(3θ1) < α).

We perform now a dual construction of the one we made in the proof ofClaim 1. Let C = {v(3θ) ≤ α : θ ∈ F} and define for each c ∈ C

uc = max{u(θ) : θ ∈ F, v(3θ) = c}.

Note that u0 = 0 by ##5 above, and uα = 1 because u(φ) = 1. Define anascending sequence 0 = c0 < c1 < .... in C as follows:

12

c0 = v(3⊥) = 0c1 = min{c ∈ C : c > c0 and uc > uc0}c2 = min{c ∈ C : c > c1 and uc > uc1}etc.

Choose φi such that uci = u(φi), ci = v(3φi). Clearly, 0 = uc0 < uc1 < ....By finiteness of F the sequence of the ci ends necessarily with cN = α,because ci = v(3φi) < α implies uci = u(φi) < 1 = uα by ##1 above andthus the existence of ci+1 ≤ α. This means also that ucn = 1.

Fix ε > 0 such that α− ε > cN−1, and further define (taking max ∅ = 0)

qN−1 = max{α− ε,max{v(�θ) : v(�θ) < cN}}qi = max{ci,max{v(�θ) : v(�θ) < ci+1}}, for i < N − 1.

Then we have:

0 = c0 ≤ q0 < c1 ≤ q1 < ....cN−1 ≤ α− ε ≤ qN−1 < cN = α0 = uc0 < uc1 < ..... < ucN = 1.

Choose g : [0, 1]→ [0, 1] to be any strictly increasing function such that

g(0) = 0g[(uci , uci+1 ]] = (qi, ci+1] for i < N − 1g[(ucN−1 , 1)] = (qN−1, α)g(1) = 1

Then g is a Heyting homomorphism and the valuation w = g ◦ v satisfiesw(φ) = w(Σ ∪ TG23) = 1; thus w ∈ W. It remains to show that SF vw ≥α− ε. Indeed, we have:

i) If v(3θ) ≥ α then trivially (w(θ) ⇒ v(3θ)) ≥ α. In particular,(w(φ)⇒ v(3φ)) = (1⇒ v(3φ)) = α.

ii) If v(3θ) < α then w(θ) ≤ v(3θ). To see this consider cases. First:u(θ) ∈ (uci , uci+1) for some i (recall u(θ) < 1 by ##1), then w(θ) ∈ (qi, ci+1].As u(θ) > uci and ci+1 = v(3φi+1) is the smallest v(3ψ) with u(ψ) > ucithen v(3θ) ≥ ci+1 ≥ w(θ). Second: u(θ) = 0, then w(θ) = 0 and v(2θ) = 0by ##4.

iii) If v(�θ) ≥ α then (v(�θ)⇒ w(θ)) > α−ε, because v(�θ) > cN−1 =v(3φN−1) which implies u(θ) > u(φN−1) = ucN−1 by ##2. Therefore,w(θ) > qN−1 ≥ α− ε by definition.

iv) v(�θ) < α then v(�θ) ≤ w(θ). To see this notice that ci ≤ v(�θ) ≤qi < ci+1 for some i and consider cases. First: v(�θ) = ci = v(3φi) then,by ##3, uci = u(φi) ≤ u(θ). Therefore ci ≤ w(θ). That is, v(�θ) ≤ w(θ).Second: ci < v(�θ) then uci < u(θ), by ##2, and by definition qi ≤ w(θ),which shows again v(�θ) ≤ w(θ).

13

From (i,ii) we have: infθ∈F {w(θ) ⇒ v(3θ)} = α, and from (iii,iv):infθ∈F {v(�θ)⇒ w(θ)} ≥ α− ε. Hence, SF vw ≥ α− ε. �

Lemma 3.2 (Weak completeness) For any finite theory T and formulaφ in L�3, T |=GK φ implies T ⊢G�3

φ.

Proof: Assume T is finite and T ⊢G�3φ. Then T, ThG�3 ⊢G φ by Lemma

2.2 and thus there is, by Proposition 2.1, a Godel valuation v : V ar ∪X →[0, 1] such that v(φ) < v(T ) = v(ThG�3) = 1. Let F be a finite fragmentcontaining T ∪ {φ}, then v is a world of the canonical model M∅,F and byLemma 3.1, eF (v, T ) = v(T ) = 1 and eF (v, φ) = v(φ) < 1. Thus T |=GK φ.�

To prove strong completeness we utilize compactness of first order clas-sical logic and the following result of Horn:

Lemma 3.3 ( [20], Lemma 3.7) Any countable linear order (P,<) may beembedded in (Q ∩ [0, 1], <) preserving all joins and meets existing in P .

Theorem 3.1 (Strong completeness) For any countable theory T andformula φ in L�3, T ⊢G�3

φ if and only if T |=GK φ.

Proof: One direction of the equivalence follows from Theorem 2.1 (sound-ness). For the other direction, assume T 0G�3

φ and consider the firstorder theory T ∗ with two unary relation symbols W,P, a binary relationsymbol <, three constant symbols 0, 1, c, two binary function symbols ◦, S,and a unary function symbol fθ for each θ ∈ L�3(V ), where V is the set ofpropositional variables occurring in formulas of T, and having for axioms:

∀x¬(W (x) ∧ P (x))∀x(W (x) ∨ ¬W (x))“(P,<) is a strict linear order with minimum 0 and maximum 1”∀x∀y(W (x) ∧W (y)→ P (S(x, y)))∀x∀y(P (x) ∧ P (y)→ (x ≤ y ∧ x ◦ y = 1) ∨ (x > y ∧ x ◦ y = y))∀x(W (x)→ f⊥(x) = 0)for each θ, ψ ∈ L�3 the sentences:∀x(W (x)→ P (fθ(x)))∀x(W (x)→ fθ∧ψ(x) = min{fθ(x), fψ(x)})∀x(W (x)→ fθ→ψ(x) = (fφ(x) ◦ fψ(x))∀x(W (x)→ f�θ(x) = infy(S(x, y) ◦ fθ(y))∀x(W (x)→ f3θ(x) = supy(min{S(x, y), fθ(y)})for each δ ∈ T the sentence: fδ(c) = 1

14

finally, W (c) ∧ (fφ(c) < 1).

For each finite part t of T ∗ let F be a finite fragment of L�3 containing{θ : fθ occurs in t}. Since F ∩ T 0G�3

φ by hypothesis, then, by weakcompleteness, there is a GK-model M∅,F = (W,SF , eF ) and a ∈W such thateF (a, θ) = 1 for each θ ∈ F ∩ T and eF (a, φ) < 1. Therefore the first orderstructure (W⊔[0, 1],W, [0, 1], <, 0, 1, a,⇒, SF , fθ)θ∈L�3

, with fθ :W → [0, 1]defined as fθ(x) = eF (x, θ), is clearly a model of t. By compactness of firstorder logic and the downward Lowenheim theorem, T ∗ has a countable modelM∗ = (B,W,P,<, 0, 1, a, ◦, S, fθ)θ∈L�3

. Using Horn’s lemma [20], (P,<)may be embedded in (Q ∩ [0, 1], <) preserving 0, 1, and all suprema andinfima existing in P ; therefore, we may assume without loss of generalitythat the ranges of the functions S and fθ are contained in [0, 1]. Then, it isstraightforward to verify that M = (W,S, e), where e(w, θ) = fθ(w) for allw ∈ W and θ ∈ L�3(V ), is a GK-model with a distinguished world a suchthat M |=a T, and M |=a φ. Hence, T |=GK φ. �

For normal theories we obtain a finer result:

Theorem 3.2 If T is a countable normal theory there is GK-model MT

such that for any φ : T ⊢G�3φ if and only if MT |= φ.

Proof: Assume T 0G�3φ. Then, by Lemma 3.1, for each finite fragment F of

L�3(V ) containing φ the canonical model MT,F is such that MT,F |= T ∩Fand MT,F |= φ. Add to the theory T ∗ in the proof of Theorem 3.1 thesentence ∀x(W (x) → fδ(x) = 1) for each δ ∈ T . Then by the previousobservation each finite part of T ∗ has a model. Arguing as in the quotedproof, we obtain a GK-model Mφ = ⟨Wφ, Sφ, eφ⟩ such that eφ(w, T ) = 1for all w ∈ Wφ and e(wφ, φ) < 1. Define now MT = (W,S, e) where W =⨿φ{Wφ : T 0G�3

φ}, Sww′ = Sφww′ if w,w′ ∈ Wφ and 0 otherwise, and

e(w, p) = eφ(w, p) for w ∈ Wφ. It is easily verified by induction on thecomplexity of θ that e(w, θ) = eφ(w, θ) for any w ∈Wφ. Thus, MT |= T andhence T ⊢G�3

φ implies MT |= φ by soundness; reciprocally, if T ⊢G�3φ

then e(wφ, φ) = eφ(wφ, φ) < 1 by construction, and thus MT |= φ. �We can not expect similar results for uncountable theories by the obser-

vation after Proposition 2.1. In fact,

Proposition 3.1 There is no single linearly ordered Heyting algebra H giv-ing strong completeness with respect to HK-models for theories of arbitrarypower, even in Godel-Dummett logic.

15

Proof: Assume otherwise, then H would be infinite (by the known Godelargument). Let κ be a cardinal greater than |H| and consider the theoryT = {(pβ → pα) → q : α < β < κ}. Then T |=HK q, because v(T ) = 1with v(q) < 1 would imply v(pβ → pα) < 1, and thus v(pα) < v(pβ) forall α < β < κ, yielding a subset of H of power κ, which is impossibleby hypothesis. On the other hand, T 0G23 q. Otherwise, we would have∆ ⊢G23 q and thus ∆ |=HK q, for some finite set ∆ = {(pαi+1 → pαi)→ q :1 ≤ i < n}, which is impossible because the valuation v(pαi) = hi, v(q) = h,where h1 < h2 < ... < hn+1 < h < 1 makes v((pαi+1 → pαi) → q) = 1 for1 ≤ i < n. �

4 Optimal models, modal axioms

The notions and results in this section make sense and hold for HK-modelswhere is H any complete Heyting algebra. Thus we state and prove themin this general framework.

Call a HK-frame M = ⟨W,S⟩ reflexive if Sxx = 1 for all x ∈ W ,transitive if Sxy · Syz ≤ Sxz for all x, y, z ∈ W, symmetric if Sxy = Syxfor all x, y ∈W, and euclidean if Sxy · Sxz ≤ Syz for all x, y, z ∈W.

Let Ref , Trans, Symm, and Euclid denote, respectively, the classesof HK- models over frames satisfying, respectively, each one of the aboveproperties. These are the fuzzy versions of the corresponding properties ofclassical frames, classically characterized by the following pairs of modalschemes:

T� 2φ→ φ T3 φ→ 3φ reflexivity4� 2φ→ 22φ 43 33φ→ 3φ transitivityB1 φ→ �3φ B2 3�φ→ φ symmetryE1 3φ→ �3φ E2 3�φ→ �φ euclidean property

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Lemma 4.1 i) T� and T3 are valid in Ref . ii) 4� and 43 are valid inTrans. iii) B1 and B2 are valid in Symm. iv) E1 and E2 are valid in Euclid.

Proof: i) In reflexive models, e(x,2φ) ≤ (Sxx ⇒ e(x, φ)) = e(x, φ) =Sxx · e(x, φ) ≤ e(x,3φ) for any x. Thus e(x,2φ→ φ) = 1 = e(x, φ→ 3φ).

ii) In transitive models, e(x,2φ) · Sxy · Syz ≤ [(Sxz ⇒ e(z, φ)) · Sxz] ≤e(z, φ) for all x, y, z. Hence, e(x,2φ) · Sxy ≤ (Syz ⇒ e(z, φ)) and thuse(x,2φ) · Sxy ≤ e(y,2φ). Therefore, e(x,2φ) ≤ (Sxy ⇒ e(y,2φ)) for ally and thus e(x,2φ) ≤ e(x,22φ) which yields 4�. Also Sxy · Syz · e(z, φ)≤ Sxz · e(z, φ) ≤ e(x,3φ). Hence, Syz · e(z, φ) ≤ (Sxy ⇒ e(x,3φ)) and

16

thus e(x,3φ) ≤ (Sxy ⇒ e(x,3φ)). Therefore, Sxy · e(x,3φ) ≤ e(x,3φ))for all y and thus e(x,33φ) ≤ e(x,3φ) which gives 43.

iii) In symmetric models, Sxy · e(x, φ) = Syx · e(x, φ) ≤ e(y,3φ) forall x, y. Then e(x, φ) ≤ (Sxy ⇒ e(y,3φ)) and thus e(x, φ) ≤ e(x,�3φ))which is B1. Moreover, e(y,�φ) ≤ (Syx ⇒ e(x, φ)); then Sxy · e(y,�φ) =Syx · e(y,�φ) ≤ e(x, φ) and thus B2 follows.

iv) In euclidean models, Sxy · e(y, φ) ·Sxz = Szy · e(y, φ) ≤ e(z,3φ) forall x, y, z. Then Sxy · e(y, φ) ≤ (Sxz ⇒ e(z,3φ)) and E1 follows. Similarly,Sxy·e(y,�φ)·Sxz ≤ Syz·(Syz ⇒ e(z, φ)) ≤ e(z, φ)), and thus Sxy·e(y,�φ)≤ (Sxz ⇒ e(z, φ)), from which E2 follows. �

To extend the completeness theorem to the [0,1]-valued analogues of theclassical bi-modal systems T, S4, S5, we introduce a particular kind of GK-model, which advantage is that the many-valued counterpart of classicalstructural properties of frames may be characterized in them by the validityof the corresponding classical schemes.

Definition 4.1 Given a HK -model M = (W,S, e), define a new accessi-bility relation S+xy = S�xy· S3xy, where S�xy = infφ∈L�3

{e(x,2φ) ⇒e(y, φ)} and S3xy = infφ∈L�3

{e(y, φ) ⇒ e(x,3φ)}. Call M optimal ifS+ = S.

The following lemma shows that any model is equivalent to an optimalone.

Lemma 4.2 (W,S+, e) is optimal. If e+ is the extension of e in this modelthen e+(x, φ) = e(x, φ) for any φ ∈ L�3.

Proof: The first claim follows from the second (which implies S++ = S+),and the second is proven by induction on the complexity of formulas. Theonly non trivial step is that of the modal connectives. Notice first that Sxy ≤S+xy, because e(x,2φ) ≤ (Sxy ⇒ e(y, φ)) and Sxy · e(y, φ) ≤ e(x,3φ)for any φ; thus Sxy ≤ (e(x,2φ) ⇒ e(y, φ)), (e(y, φ) ⇒ e(x,3φ)). Now,assume e+(y, φ) = e(y, φ) for all y, then by the previous observation and theinduction hypothesis: e+(x,�φ) = infy{S+xy ⇒ e+(y, φ)} ≤ infy{Sxy ⇒e(y, φ)} = e(x,�φ). But S+xy ≤ (e(x,�φ) ⇒ e(y, φ)) by definition ofS+ and thus e(x,�φ) ≤ (S+xy ⇒ e(y, φ)) = (S+xy ⇒ e+(y, φ)) whichyields e(x,�φ) ≤ e+(x,�φ). Similarly, by the induction hypothesis and thefirst observation, e+(x,3φ) = supy{S+xy · e(y, φ)} ≥ supy{Sxy · e(y, φ)} =e(x,3φ), and by definition S+xy ≤ (e(y, φ) ⇒ e(x,3φ)). Thus S+xy ·e+(y, φ) = S+xy · e(y, φ) ≤ e(x,3φ) which yields e+(x,3φ) ≤ e(x,3φ). �

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Proposition 4.1 An optimal HK-model is: i) reflexive if and only if it val-idates the schemes T�+T3, ii) transitive if and only if it validates 4�+43,iii) symmetric if and only if it validates B1+B2, iv) euclidean if and only ifit validates E1+E2.

Proof: i) If T� and T3 hold, Sxx = infφ{e(x,�φ → φ)} · infφ{e(x, φ →3φ)} = 1, by optimality.

ii) S�xy · S�yz ≤ (e(x,��φ) ⇒ e(y,�φ)) · (e(y,�φ) ⇒ e(z, φ)) ≤(e(x,��φ)⇒ e(z, φ)) ≤ (e(x,�φ)⇒ e(z, φ)), the last inequality holding by4�. Similarly, S3xy · S3yz ≤ (e(y,3φ)⇒ e(x,33φ)) · (e(z, φ)⇒ e(y,3φ))≤ (e(z, φ)⇒ e(x,33φ)) ≤ (e(z, φ)⇒ e(x,3φ)), the last inequality holdingby 43. Hence, Sxy · Syz ≤ (e(x,�φ) ⇒ e(z, φ)) · (e(z, φ) ⇒ e(x,3φ) byoptimality. Taking meet over φ in the right we get transitivity.

iii) Since S�xy ≤ (e(x,�3φ) ⇒ e(y,3φ)) ≤ (e(x, φ) ⇒ e(y,3φ)) byB1, then taking meet over φ, we obtain S�xy ≤ S3yx. Similarly, S3yx≤ (e(x,�φ)⇒ e(y,3�φ)) ≤ (e(x,�φ)⇒ e(y, φ)) by B2, and then S3yx ≤S�xy. From this, S3xy = S�yx, and thus Sxy = Syx.

iv) Assuming E1, S3xz ≤ (e(z, φ)⇒ e(z,3φ)) ≤ (e(z, φ)⇒ e(z,�3φ))for any formula φ, and combining this with S�xy ≤ (e(x,�3φ)⇒ e(y,3φ)),we obtain S�xy · S3xz ≤ (e(z, φ) ⇒ e(z,3φ)). Similarly, assuming E2,S3xy ≤ (e(y,�φ) ⇒ e(x,3�φ)) ≤ (e(y,�φ) ⇒ e(x,�φ)), and combin-ing this with S�xz ≤ (e(x,�φ) ⇒ e(z, φ)), we obtain S3xy · S�xz ≤(e(y,�φ) ⇒ e(z, φ)). Multiplying the obtained inequalities we get: Sxy ·Sxz ≤ (e(y,�φ)⇒ e(z, φ)) · (e(z, φ)⇒ e(z,3φ)), which yields Sxy · Sxz ≤Syz by optimality. �Remark. Another relevant property of classical Kripke frames is seriality:∀x∃ySxy = 1, characterized (classically) by any of the axioms 3⊤ or ¬�⊥.We have not been able to characterize this property in GK-frames. How-ever, its fuzzy version: ∀x supy∈W Sxy = 1, is readily seen to be equivalentin arbitrary HK-frames to the validity of 3⊤, while the axiom ¬�⊥ char-acterizes only the weaker condition ∀x∃ySxy > 0.

5 Godel analogues of classical bi-modal systems

Lemma 4.2, in conjunction with Proposition 4.1 and Theorem 3.1, implystrong completeness of any combination of axiom pairs in Table 3, withrespect to GK-models over frames satisfying the associated structural prop-erties. In particular, we obtain completeness for the Godel analogues of theclassical modal systems T , S4 and S5:

18

GT�3 := G�3 +T� +T3

GS4�3 := G�3 +T� +T3 + 4� + 43GS5�3 := G�3 +T� +T3 + 4� + 43 +B1 +B2

These systems may be seen to be equivalent, respectively, to the purelyintutionistic modal logics IT, IS4, IS5 = MIPC ([14], [28], [4]) plus theprelinearity scheme. We let the reader consider other relevant combinations.Recall that strong completeness refers here to entailment from countabletheories.

Theorem 5.1 GT�3 is strongly complete for |=GK∩Ref .GS4�3 is strongly complete for |=GK∩Ref∩Trans .GS5�3 is strongly complete for |=GK∩Ref∩Trans∩Symm.

Proof: If T |=GK∩Ref φ then T |=GK∩Optimal∩Ref φ. Thus T + {T�,T3}|=GK∩Optimal φ by Proposition 4.1, and T + {T�,T3} |=GK φ by Lemma4.2. Therefore, T + {T�,T3} ⊢G�3

φ by 3.1, which implies T ⊢GT�3φ. The

proofs of the other two cases are similar. �We focus on the system GS5�3 which may be considerably simplified

because the symmetry axioms B1 and B2 imply the inter-deducibility ofeach pair {FS1,FS2}, {T�,T3} and {4�,43}, and B2 deduces F3. Moreover,in the presence of {T�,T3}, the euclidean axioms {E1,E2} are equivalent to{4�,43}+{B1,B2}; therefore, we are left with the modal axioms:

�(φ→ ψ)→ (�φ→ �ψ)3(φ ∨ ψ)→ (3φ ∨3ψ)�(φ→ ψ)→ (3φ→ 3ψ)2φ→ φφ→ 3φ3φ→ �3φ3�φ→ �φ

GS5�3 presents some features which distinguish it from the weaker systemsG�3,GT�3 and GS4�3. The uni-modal fragments of the latter logics havesimple axiomatizations while axiomatizations for the uni-modal fragmentsof GS5�3 are unknown. The �-fragments of the weaker systems are char-acterized by their accessibility-crisp models, as shown in [6], but this is notthe case for the �-fragments of GS5�3, as the following example illustrates.

Example. The formula �(�φ∨ψ)→ (�φ∨�ψ) is not a theorem of GS5�but it is valid in any accessibility-crisp model of GS5�. The first claim isgranted by the following two worlds model:

19

{p = 12 , q = 1} u

1

←→12

v1

{p = 12 , q = 0}

in which the reader may verify that e(u,�(�p∨q)) = 1 and e(u,�p∨�q) =12 . To verify the second claim notice that if (W,S, e) ∈ Ref ∩Trans∩Symmhas crisp accessibility S, this defines a classical equivalence relation ∼ in Wand thus e(x,�θ) = infy{Sxy ⇒ e(y, θ)} = infy∼x{e(y, θ)} for any formulaθ. Therefore, e(x,�(�φ ∨ ψ)) = infy∼x{infz∼y e(z, φ) g e(y, ψ)}. But αy =e(y,�φ) = infz∼y e(z, φ) is identical to αx for all y ∼ x because {z : z ∼y} = {z : z ∼ x}; hence, e(x,�(�φ ∨ ψ)) = infy∼x{αx g e(y, ψ)} = αx ginfy∼x{e(y, ψ)} = e(x,�φ ∨�ψ) by distributive properties of [0,1].

In the classical setting, S5 is characterized by Kripke models with uni-versal accessibility relation; that is, Sxy = 1 for all x, y. This can not be thecase for GS5�3 or its �-fragment due to the previous example, nor is it thecase for the 3-fragment because ¬¬3φ → 3¬¬φ holds in all accessibility-crisp models but fails at the world v in the model displayed in the previousexample (φ := q). However,

Theorem 5.2 GS5∗�3 := GS5�3 + {�(�φ ∨ ψ) → (�φ ∨ �ψ)} is stronglycomplete for |=GK∩Universal.

Proof: Weak completeness with respect to GK-models over universal framesis shown by Hajek in ([19]) for the deductively equivalent system S5(G). Thismay be extended to strong completeness with respect to countable theoriesas in the proof of Theorem 3.1. �

Clearly, the new scheme yields completeness with respect to accessibilitycrisp models of GS5�3, but it is not even valid in the accessibility crispmodels of the weaker Godel modal logics.

6 The algebraic connection

As an algebrizable deductive logic, G�3 has a unique algebraic semanticsgiven by the variety of bi-modal Godel algebras, those of the form A =(G, I,K) where G is a Godel algebra and I and K are unary operations inG satisfying the identities:

I(a · b) = Ia · Ib K(ag b) = KagKbI1 = 1 K0 = 0Ka→ Ib ≤ I(a→ b) K(a→ b) ≤ Ia→ Kb

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This means that G�3 is complete with respect to valuations v : V ar → A inthese algebras, when they are extended to L�3 interpreting � and 3 by Iand K, respectively.

Similarly, GT�3, GS4�3, and GS5�3 have for algebraic semantic thesubvarieties of bi-modal Godel algebras determined by the pairs of identitiesin the following table corresponding to their characteristic axioms:

Ia ≤ a a ≤ Ka reflexivityIa ≤ IIa Ka ≤ KKa transitivitya ≤ IKa KIa ≤ a symmetryKa→ IKa KIa ≤ Ia euclidean property

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Notice that the algebraic models of GS4�3 are just the bi-topological pseudo-Boolean algebras of Ono [24] with a Godel algebra as underlying Heytingalgebra, and the algebraic models of GS5�3 are the the monadic Heytingalgebras of Monteiro and Varsavsky [23] with a Godel underlying algebra.

Example. As we have noticed, there is no finite counter-model for theformula 2¬¬p → ¬¬2p in Godel-Kripke semantics. However, the algebraA = ({0, a, 1}, I,K), where {0 < a < 1} is the three elements Godel algebraand I1 = 1, Ia = I0 = 0, K1 = Ka = 1, K0 = 0 is a bi-modal Godel algebra(actually a monadic Heyting algebra) providing a finite counterexample tothe validity of this formula by means of the valuation v(p) = a, as the readermay verify.

We may associate to each Godel-Kripke frame F = (W,S) a bi-modalGodel algebra [0, 1]F = ([0, 1]W , IF ,KF ) where [0, 1]W is the product Godelalgebra, and for each map f ∈ [0, 1]W :

IF (f)(w) = infw′∈W

(Sww′ ⇒ f(w′))

KF (f)(w) = supw′∈W

(Sww′ · f(w′))

Theorem 6.1 [0, 1]F is a bi-modal Godel algebra, and there is a one toone correspondence between Godel-Kripke models over F , and valuationsv : V ar → [0, 1]F given by the adjunction:

V ar ×W e→ [0, 1]

V arve→ [0, 1]W , ve(p) = e(−, p)

so that ve(φ) = e(−, φ) for any formula φ. Moreover, the transformationF 7−→ [0, 1]F sends reflexive, transitive, symmetric, and euclidean Godel-Kripke frames, respectively, into bi-modal algebras satisfying the correspond-ing identities.

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Proof: The verification of the identities that IF , KF must satisfy in eachcase is routine and the induction on formulas showing ve(φ)(w) = e(w,φ) isstraightforward. �

Reciprocally, utilizing our strong completeness theorem for normal the-ories (Theorem 3.2), we may associate to each countable bi-modal Godelalgebra A a GK-frame F such that A may be embedded in the associatedalgebra [0, 1]F , and to each algebraic valuation v in A corresponds a GK-model over F validating the same formulas as v. However, the constructionis not canonical.

Theorem 6.2 For any countable bi-modal Godel algebra A there is Godelframe F = (W,S) such that:i) Any pair of identities in (4) which is valid in A is valid in [0, 1]F .ii) A is embeddable in the algebra [0, 1]F .iii) For any valuation v : V ar → A there exists ev : W × V ar → [0, 1] suchthat (W,S, ev) |= φ if and only if v(φ) = 1, for any sentence φ.

Proof: Fix a valuation η into A with onto extension η : L�3 → A and letT = {φ : η(φ) = 1}. Then T is a normal theory deductively closed and forthe model MT = (W,S, e) of Theorem 3.2 we have MT |= φ if and only ifη(φ) = 1. Without loss of generality we may assumeMT is optimal (Lemma4.2). Set F := (W,S), then (i) holds by Proposition 4.1 and the last claimof Theorem 6.1. To see (ii) notice that, by the same theorem, e inducesa bi-modal Godel valuation ve : V ar → [0, 1](W,S), ve(p) = e(−, p) suchthat ve(φ) = e(−, φ) = 1 ∈ [0, 1]W if and only if η(φ) = 1. This meansthat the extension ve : L�3 → [0, 1](W,S) factors injectively through η; thatis, ve = δ ◦ η for an injective homomorphism of bi-modal Godel algebrasδ : A → [0, 1](W,S). Finally, to show (iii), pick v : V ar → A, then δ ◦ v isa valuation into [0, 1](W,S) which induces, by Theorem 6.1, a GK-valuationev :W ×V ar → [0, 1] such that ev(w,φ) = δ(v(φ))(w). As δ is one to one wehave that v(φ) = 1 if and only if δ(v(φ)) = 1 ∈ [0, 1]W ; that is, ev(w,φ) = 1for all w, which means (W,S, ev) |= φ. �

Applying parts (i) and (ii) of the previous theorem to the free algebrasof countable rank we obtain:

Corollary 6.1 The variety of bi-modal Godel algebras is generated by analgebra of the form [0, 1]

(W,S). A similar result holds for the subvarieties

determined by any combination of identity pairs in (4).

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As we defined [0, 1](W,S) we may define, similarly, bi-modal algebrasC(W,S) where C is any complete chain, and obtain from Theorem 6.2, uti-lizing ultraproducts and the Dedekind-MacNeille completion:

Theorem 6.3 For any bi-modal Godel algebra A there is a complete chainC and a GK-frame (W,S) such that A is embeddable in C(W,S). Moreover,the latter algebra satisfies the same identity pairs in (4) as A.

7 Afterword

Our objective of axiomatizing the main bi-modal fuzzy logics under theGodel-Kripke interpretation is fully achieved, and it is not difficult to ex-tend this to languages enriched with sets of truth-constants along the linesof similar results for the uni-modal fragments in [6]. But some particularaxiomatizability problems are left open in this paper. We have empha-sized already the lack of an axiomatization for the uni-modal fragments ofGS5�3. Another problem is the axiomatizability of validity in accessibility-crisp models of each logic considered, having effective solutions for the frag-ments G�, GT�, and GS4� (the logic themselves [6]), the fragment G3 (addthe rule (φ→ ψ)∨ θ/(3φ→ 3ψ)∨3θ, Metcalfe and Olivetti [22]), and thelogic GS5�3 (the extension GS5∗�3 introduced in Theorem 5.2).

The question on the decidability and complexity of G�3 and its ex-tensions is also left unanswered since these logics do not have the finitemodel property under GK-semantics. However, the uni-modal fragmentsG�, G3,GT3, and GS43 are known to be decidable, the first two by resultsof Metcalfe and Olivetti [22] who show they are PSPACE-complete, and thelast three because they do have the finite model property (see [6]). As in[22], we may utilize a double negation interpretation of the classical modallogics into their Godel counterparts to show that GS5�3 is co-NP-hard andthe other logics considered here are PSPACE-hard.

It has been noticed throughout the paper that most results reported,excepting deductive completeness, hold for HK-models where H is an ar-bitrary complete Heyting algebra. It is reasonable to expect that validityin HK-models is axiomatized by IK + LH , where IK is Heyting calculusplus the set of modal axioms of G�3 and LH denotes an axiomatization ofH-valued propositional logic. However, our completeness proof with respectto GK-models does not shed light on this hypothesis because it dependsheavily on the linear and homogeneous character of [0,1].

To finish, we must thank the constructive criticism of an anonymousreferee which lead to the improvement of this paper.

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